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Year 7AC9M7M04

Angle Relationships

Identify corresponding, alternate and co-interior relationships between angles formed when parallel lines are crossed by a transversal; use them to solve problems and explain reasons.

When a line crosses two parallel lines, the angles come in matching pairs. Learn the four names and most questions solve themselves.

Method

Angles on a straight line
a+b=180∘a + b = 180^\circ
They make a half turn.
Angles at a point
a+b+c+d=360∘a + b + c + d = 360^\circ
A full turn.
Vertically opposite
a=ba = b
The pair facing each other across a crossing are equal.
Angles in a triangle
a+b+c=180∘a + b + c = 180^\circ
Always.

Worked example

e.g. a line crosses two parallel lines. One angle is 70∘\text{a line crosses two parallel lines. One angle is } 70^\circ
  1. Mark the angle you are given70∘70^\circ
    A transversal crossing two parallel lines, one angle marked 70 degrees70°Start from the angle you know
  2. Vertically opposite angles are equal. Mark its partner across the crossing70∘70^\circ
    The vertically opposite angle is also 70 degrees70°70°Vertically opposite angles are equal
  3. Angles on a straight line add to 180∘180^\circ. Fill in the other two180−70=110∘180 - 70 = 110^\circ
    The remaining angles at the top crossing are 110 degrees70°70°110°110°
  4. Corresponding angles are equal. Copy the pattern down to the second line70∘ and 110∘70^\circ \text{ and } 110^\circ
    The same pattern of angles repeats at the lower crossing70°110°70°110°Corresponding angles match

Common mistakes

WrongCo-interior angles are equal\text{Co-interior angles are equal}
RightCo-interior angles add to 180∘\text{Co-interior angles add to } 180^\circ
The two angles between the parallel lines, on the same side of the crossing line, are supplementary — not equal. They only look equal when the crossing line is at right angles.
Wronga+b=180∘ for any two anglesa + b = 180^\circ \text{ for any two angles}
RightOnly when they sit on a straight line\text{Only when they sit on a straight line}
Check the angles actually make a half turn before you use it.
WrongAssuming lines are parallel because they look it\text{Assuming lines are parallel because they look it}
RightLook for the arrowheads\text{Look for the arrowheads}
Matching arrowheads mark parallel lines. Without them you cannot use corresponding or alternate angle rules.

Practice

Find the missing angle:

1
On a straight line: 65∘ and x\text{On a straight line: } 65^\circ \text{ and } x
Answerx=115∘x = 115^\circ
2
At a point: 90∘,120∘,85∘ and x\text{At a point: } 90^\circ, 120^\circ, 85^\circ \text{ and } x
Answerx=65∘x = 65^\circ
3
Vertically opposite 42∘\text{Vertically opposite } 42^\circ
Answer42∘42^\circ
4
Triangle with 55∘ and 75∘\text{Triangle with } 55^\circ \text{ and } 75^\circ
Answer50∘50^\circ
5
Corresponding to 118∘\text{Corresponding to } 118^\circ
Answer118∘118^\circ
6
Co-interior with 118∘\text{Co-interior with } 118^\circ
Answer62∘62^\circ
7
Alternate to 37∘\text{Alternate to } 37^\circ
Answer37∘37^\circ
8
Triangle with two angles of 60∘\text{Triangle with two angles of } 60^\circ
Answer60∘60^\circ
9
On a straight line: 90∘ and x\text{On a straight line: } 90^\circ \text{ and } x
Answerx=90∘x = 90^\circ
10
At a point: 145∘,145∘ and x\text{At a point: } 145^\circ, 145^\circ \text{ and } x
Answerx=70∘x = 70^\circ
Next step
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